Z. Suppose A and B are m x n matrices. Prove that C(A) and C(B) are orthogonal subspaces of R" if and only if ATB = O. checikinnli

Q: 1. Consider the following subspaces of R³: U = {(x,x,y): x,y ER} and W = {(x,-x, 2x): x, y ≤ R} (a)…

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Q: 1. Find a basis for the vector subspace of 2 x 3 matrices given by а 6 c V := { : a + b+ c= 0, a = c…

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Q: Find the intersection of the following two affine subspaces of R4. 1 1 0 1 1 1 1 1 0 0 1 y 1 1

A: Let's find.

Q: 1. For each of the following matrices, determine a basis - and N(A"): 4 subspaces R(A"), N(A), R(A),…

A: Let us apply elementary row operations on the given Matrix A to reduce it in reduced row Echelon…

Q: b] Let W be the set of all 2 × 2 matrices A = such that Az = 0, where z = ||. Is W subspace of M22?…

A: A set W of vector space V over field F is said to be subspace of vector space V if W is itself a…

Q: For each of the following matrices, determine a basis for each of the subspaces R(AT ), N(A), R(A),…

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Q: Suppose that A is an n x m matrix of rank 3, the nullity of A is 3, and the column space of A is a…

A: Answer (6)  Suppose that A is an n × m matrix of rank 3 , the nullity of A is 3 , and the column…

Q: Let S = E M2x2; a = d and b +c = 0} be a subspace of M2x2- Then the dimension of S is equal to: 1 4

A: S=abcd∈M2×2; a=d and b+c=0 be a subspace of M2×2

Q: 5.7.5 ND Suppose that V has an inner product, and that n is in V. Show that the set Vo of all…

A: We will use the definition to prove that, V0 is a subspace of V, keeping in mind the properties of…

Q: -10 e subspace W of R" spanned by 6 and 3

A: In this question, concept of orthogonal projection is applied. Orthogonal Projection Matrix of…

Q: 2.14. Prove that two subspaces V1 and V2 are linearly independent if and only if Vịn V2 = {0}.

A: V1 and V2 are two subspace.

Q: 1. Suppose that A is an nxn non-zero, real matrix and 2 is a fixed real number. Let E ={x=R* : AT =…

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Q: Determine is Question 2 whether W = W = {(a; z { (a; 2; 6) | α; 6ER} subspace of R

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Q: Find a basis for the subspace of R3 spanned by S.S = {(4, 4, 8), (1, 1, 2), (1, 1, 1)}

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Q: Find the orthogonal complement S⊥ of the subspace S of R3 spanned by the two column vectors of the…

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Q: 2) One of the following subsets of IR2 is not subspace: a) S = {, xeR} b) s = {], = x3} %3D %3D e) s…

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Q: Suppose that A is an n x m matrix of rank 5, the nullity of A is 4, and the image of A is a subspace…

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Q: Prove that W1 = {(a1, a2, . . . , an) ∈ Fn: a1 + a2 + · · · + an= 0} is a subspace of Fn , but W2 =…

A: Prove that W1 = {(a1, a2, . . . , an) ∈ Fn: a1 + a2 + · · · + an= 0} is a subspace of Fn , but W2 =…

Q: For an m x n matrix A, the column space is a subspace of Rn. True False

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Q: Solve: (a) If a 7 x 9 matrix A has rank 5, what are the dimensions of the four fundamental…

A: If the order of the matrix is m×n and rank of the matrix is r. Then dimension of column space =r…

Q: 1. Which of the following sets of matrices form a subspace of M22? (assume a, b range across the…

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Q: If Sz is a subspace of R" of dimension 3, then there cannot exist a subspace S, of R* such that Si c…

A: This staement is true.

Q: 2) Find all the invariant subspaces of the following matrix 0. 1 0 1 1 1

A: Given matrix is, Let, A=010010111 Now, calculating the eigen value of the given matrix.…

Q: (1) Suppose U and W are both five-dimensional subspaces of R°. Prove that UnW + {0}.

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Q: If W is a 2 x 2 matrices A with trace equal to zero is a subspace of M2,2. Prove that span X = W.…

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Q: Let W be the set of all 3 x 3 matrices whose trace is zero. Show that S is a subspace of M33.

A: Let W be the set of all 3×3 matrices whose trace is zero. Show that W is a subspace of M33.

Q: Which of the following must be true? The set of all n x matrices having trace equal to zero is a…

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Q: The set {(x,3): x is in R)} is a subspace of R2

A: The given set is x,3: x is in R. It is known that any subspace should contain identity element.

Q: and T are subspaces of V, then SUT

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Q: Let A be an mxn matrix. Prove that every vector x in R" can be written in the form x = p+u, where p…

A: Let A be an m×n matrix.Prove: for evert vector x∈ℝn can be written in the form x=p+u, where p is in…

Q: Let B be a fixed mxn matrix and let S be the set of nxp matrices where S={A|BA=0}. Determine if S is…

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Q: 2. Determine which of the following is a subspace of R". (a) {(x1,x2,...X) 1 x = 0} (b) {(x1,…

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Q: Show that S = {(a,0, b)"|a,b E R} forms a subspace of R³.

A: The solution are next step

Q: Prove that W1={ (al, a2, ..., an)e Fn: al+a2+ +an=0} is a subspace of Fn, but W2= { (al, a2, ...,…

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Q: Suppose W is a set of 2 x 2 matrices A with trace A = 0 , and is a subspace of M2,2 Consider the…

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Q: Suppose that A is a fixed 2x2 matrix. Let U = {X ∈ M22 | AX = XA} Question: Prove that U is a…

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Q: - Let V = R and W = {(a, b, c) E V \a + b = c}. Is W a subspace of V? If so, what is its dimension?

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Q: Let S = {[" e Mzxzi a = d and b + c = 0} be a subspace of M2x2- Then the dimension of S is equal to:…

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Q: If A is an mxn matrix, then the row space of A is a subspace of: O A. Rmn B. R". O C. R". n-m

A: Given that A is an m×n  matrix, then the row space of A is a subspace of :

Q: If V = M2x2(R) and W consists of those elements of V in which at least one entry of the matrix is…

A: That's easy. Have a great day!!!

Q: An m×n matrix A is called upper triangular if all entries lying below the diagonal entries are zero,…

A: Proof:

Q: Find a basis for the subspace of R3 spanned by S.S = {(2, 3, −1), (1, 3, −9), (0, 1, 5)}

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Q: Given A be a 2 x 2 matrix with the form listed below: A = [0 a          b 0]  , where V = M2,2.…

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Q: Let A be an m×n matrix. Find all b such that the set of solutions to Ax = b is a subspace and prove.

A: Given: Let A be an m×n matrix.  The objective is to find that the set of solutions to Ax =b is a…

Q: If A is an m × n matrix and b #0 vector in R", then the solution set of Ax=b is a subspace of R". O…

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Q: 2. Determine if the set of all 2 x 2 matrices where tr(A) = 0 is a subspace of the vector space,…

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Q: If y is not in a subspace W, then y - ŷ is zeo. Select one: True False

A: If y is in subspace W, then the orthogonal projection of y onto W (y') is y itself.  And in that…

Q: Let A and B be m x n matrices. Show that V = {x e R" : Ax = Bx} is a subspace of R".

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Q: Which of the following sets are subspaces of Rª ? I. The solution set of the system Ar = 0, for a 4…

A: Since you have asked multiple questions, we will solve the first question for you. If you want any…

Q: Prove that is U1 and U2 are subspaces of V then dim(U1 + U2) dim(U1)+dim(U2)-dim(U1 N U2). Using…

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Linear Algebra

In mathematics, problems can be solved by the arithmetic operators like addition, subtraction, multiplication, and division. In algebra, we represent the numbers by any letter called a variable. If these variables are of degree 1, then it is studied under linear algebra. The first-degree equations involving algebraic expressions are called linear equations. That is, if you represent it by drawing a graph you will get a straight line.